ABOUT SOME INFINITE FAMILY OF 2-BRIDGE KNOTS AND 3-MANIFOLDS

We construct an infinite family of 3-manifolds and show that these manifolds have cyclically presented fundamental groups and are cyclic branched coverings of the 3-sphere branched over the 2-bridge knots ( +1)2 or ( +1)1, that are the closure of the rational (2 −1)/( −1)–tangles or (2 −1)/ –tangles, respectively.

1. Introduction. The purpose of this paper is to investigate the connection between cyclically presented groups and cyclic branched coverings of S 3 branched over knots or links. This kind of works can be found in many papers (cf. [2,4,7,8,9,12,13]).
Let F n = x 1 ,...,x n | be the free group of rank n and η : F n → F n be the automorphism of order n such that η(x i ) = x i+1 , i = 1,...,n, where the indices are taken mod n. Then for a reduced word w ∈ F n , the cyclically presented group G n (w) is given by G n (w) = x 1 ,...,x n | w, η(w),...,η n−1 (w) . (1.1) A group G is said to have a cyclic presentation if G is isomorphic to G n (w) for some n and w. Let be a knot in the 3-sphere S 3 . We will say that a 3-dimensional manifold M is an n-fold cyclic branched covering of the knot if M is an n-fold cyclic branched covering of S 3 branched over the knot (see [1,14]). In other words, M is the covering of the orbifold (n) with underlying space S 3 and the singular set the knot . In this case, the fundamental group of the manifold has a cyclic automorphism and the split extension is the group of the orbifold (n). So it is interesting to find a cyclic presentation for the fundamental group of the manifold, corresponding to this cyclic covering.
(1.2) (ii) M n is hyperbolic for n > 3 and Euclidean for n = 3.
The connection between the manifolds M n and knot theory was mentioned in [9]. Actually it was shown that (iii) M n is the n-fold cyclic branched covering of the knot 4 1 . Hence the works for the Fibonacci manifolds constitute the most beautiful examples of the connection between cyclically presented groups and cyclic branched coverings of knots and links. Actually for the construction of the Fibonacci manifold M n , a polyhedron schema was considered, that is, the boundary of 3-ball was tessellated into n triangles in the northern hemisphere, n-triangles in the southern hemisphere, 2n triangles in the equatorial zone. Then certain orientations and identifications were considered. In this paper, we will consider more general tessellation, that is, the boundary of 3-ball will be tessellated into n triangles in the northern hemisphere, -gons in the southern hemisphere, n triangles and n -gons in the equatorial zone ( ≥ 3).
For ≥ 3 and n ≥ 2, let G( , n) be a finitely generated group with the following cyclic presentation: if is even, In Section 2, we show that G( , n) can arise as a fundamental group of a closed orientable 3-manifold. In Section 3, we demonstrate that G( , n) is closely connected with the 2-bridge knot ( +1) 2 or ( +1) 1 , that is the closure of the rational (2 − 1)/( − 1)tangle or (2 − 1)/ -tangle, according as is even or odd, respectively (see [14] for notation). In Section 4, we show that the manifold obtained in Section 2 is also obtained as a 2-fold branched covering over an n-periodic knot. Finally we will have an infinite family of maximally symmetric manifolds in Section 5. Remark 1.1. In particular, if = 3, all our properties are the same as the ones for Fibonacci manifolds in [8,9].
The polyhedron P (6,3). (i) The oriented edges fall into 2n + 1 classes: x i , i = 1,...,n, where each class x i consists edges, y i , i = 1,...,n, where each class y i consists 3 edges. In this case, oriented edges from the same class carry the same label.
(ii) For each i = 1,...,n, the boundary cycle of the -gons T i and T i is ( −1)/2 with the indices taken mod n.
(iii) For each i = 1,...,n, the boundary cycle of the triangles F i and F i is with the indices taken mod n.
Note that the set of all the faces splits into pairs of faces with the same sequences of oriented boundary edges. Now we identify triangles F i with F i , and -gons T i and T i such that the corresponding oriented edges on polygons carrying the same label are identified for each i = 1,...,n. For example, if = 5 and n = 3, we have the polyhedron P (5, 3) as shown in Figure 2.1a.
The resulting complex M( , n) has one vertex, 2n 1-cells, 2n 2-cells and one 3-cell. Then we have a closed connected orientable 3-manifold M( , n) by applying a simple criterion, due to Seifert and Threlfall [15]: a complex which is formed by identifying the faces of a polyhedron will be a manifold if and only if its Euler characteristic equals zero.
For the fundamental group of M( , n) we select N as an initial point of the closed paths. Then we have the generating path classes of the fundamental group, X i = x i and Y i = y i for i = 1,...,n. By running around the boundaries of the 2n 2-cells of M( , n), we get the following relators: for i = 1,...,n, Hence the fundamental group of a manifold M( , n) is Therefore it is isomorphic to G( + 1, n). Similar arguments can be applied for the case when is even (see Figure 2.1b for the orientation and labeling of the edges of P (6, 3)).

The split extension of the group G( , n)
Proof. Let be odd. We consider a presentation for G( , n), which can be easily shown using Tieze transformation with y i (x −1 i+1 x i ) ( −1)/2 = 1 for all i = 1,...,n.
Then we see that the group G( , n) has a cyclic automorphism ρ : x i → x i+1 and y i → y i+1 of order n. We consider the split extension G( , n) of group G( , n) by the cyclic group of automorphisms generated by ρ.
With notation x = x 1 and y = y 1 , Note that ρ and x −1 ρ are conjugate. Let µ = x −1 ρ. Then x = ρµ −1 and µ n = 1. So We recall that the group For the case when is even, we can apply similar arguments.   It lies below the diagram, inside the ball whose boundary is being identified along the disc pairs F,F , and T ,T . Cancelling handles we obtain S 3 and the knot ( + 1) 1 (see Figures 3.2c, 3.2d, and 3.2e). By Theorem 3.3 (see [5,10]) we have that the orbifold ( + 1) 1 (n) (denoted ((2 − 1)/ )(n)) is hyperbolic for n ≥ 3, ≥ 3, and it is spherical for n = 2, ≥ 3.  4. The manifolds M( , n) as 2-fold coverings. In this section, we will study the topological properties of manifolds M( , n), that gives a topological approach to the studying of cyclically-presented groups G( , n). This study is analogous to the topological studying of Sieradski groups S(n) and Fibonacci groups F(2, 2n) given in [2,3,9,16].
Firstly we define a series of knots. We recall that any knot can be obtained as the closure of some braid [1]. Let p and q be coprime integers, then by σ p/q i we denote the rational p/q-tangle whose incoming arcs are ith and (i + 1)th strings. For an integer n ≥ 1 we denote by n the n-periodic knot which is the closure of the rational 3strings braid σ 2 σ    Proof. First we assume that is odd and ≥ 3. By Theorem 3.2 the manifold M( , n) is the n-fold cyclic branched covering of the 3-sphere S 3 , branched over the knot ( +1) 1 . To describe M( , n) as a 2-fold cyclic branched covering of S 3 , branched over an n-periodic knot, we will use the following construction which is analogous to [2,16] where the Fibonacci groups and the Sieradski groups were topologically studied. From Figure 4.2 we see that the orbifold ( +1) 1 (n) has a rotation symmetry of order two denoted by τ such that the axe of the symmetry is disjoint from ( +1) 1 .
It is not difficult to see that this symmetry action produces an orbifold ( + 1) 1 / τ with underlying space S 3 and the singular set the 2-component link pictured in Figure  4.3 with branch indices 2 and n. Note that the singular set of the quotient orbifold is   2, ), that is the 2-bridge link obtained as the closure of the rational (4 − 2)/ -tangle. We will denote the quotient orbifold ( + 1) 1 / τ by b(4 − 2, )(2,n).
Then we have the following covering diagram: and a sequence of normal subgroups where |Ω( , n) : G( , n)| = 2 and | G( , n) : G( , n)| = n. We describe the orbifold group Ω( , n) using the Wirtinger representation of the link group of b(4 − 2, ) in Figure 4.3. The link group has two generatorsᾱ,β and one relator of the formᾱw = wᾱ, where a word w is determined as follows: According to [6], we get the following presentation of the orbifold group Ω( , n) of the orbifold b(4 − 2, ) (2,n): where the generators α and β canonically correspond toᾱ andβ, respectively. Let us consider the group and the epimorphism defined by setting θ(α) = a and θ(β) = b. By the construction of the 2-fold covering the loop β ∈ Ω( , n) lifts to a trivial loop in G( , n). The loop α ∈ Ω( , n) lifts to a loop in G( , n) which generates a cyclic subgroup of order n. Thus it follows that  M( , n)), hence G( , n) = ker θ. Let Γ n be the subgroup of Ω( , n) given by Then we get a sequence of normal subgroups where |Ω( , n) : Γ n | = n and |Γ n : G( , n)| = 2. We recall, that the orbifold b(4 − 2, )(2,n) is spherical for n = 2, and hyperbolic for n ≥ 3. Hence the group Γ n acts by isometries on the universal covering X n , that is the 3-sphere S 3 for n = 2, and the hyperbolic space H 3 for n ≥ 3. Thus we get the orbifold X n /Γ n and the following covering diagram: n). (4.13) In this case, the second covering is cyclic and it is branched over the component with index n of the singular set of b(4 − 2, )(2,n) in Figure 4.3. But this component is the knot 1 and is trivial. So, underlying space of X n /Γ n is the 3-sphere. By the construction of the n-fold covering X n /Γ n n → b(4 − 2, )(2,n) (4.14) the loop α ∈ Ω( , n) lifts to a trivial loop in Γ n , and the loop β ∈ Ω( , n) lifts to a loop in Γ n which generates a cyclic group of order 2. Because b(4 −2, ) are 2-bridge links whose components are equivalent, we can exchange branch indices of components in Figure 4.3. Therefore, the singular set of X n /Γ n is an n-periodic knot which can be obtained as the closure of the 3-string braid σ −1 2 σ 2/( −1) 1 n , that is the knot n .
Because the branch index is equal to 2, we denote X n /Γ n = n (2).
Comparing (1) and (2), we get that the following covering diagram is commutative: If is even, we see that the orbifold ( + 1) 2 (n) has a rotation symmetry of order two denoted by τ such that the axe of the symmetry is disjoint from the knot ( +1) 2 (see Figure 4.4).

Maximally symmetric manifolds.
We recall, that the maximal possible order of a finite group G of orientation-preserving homeomorphisms of the orientable 3dimensional handlebody V g of genus g > 1 is 12(g − 1) [17], analogous to the classical 84(g − 1)-bound for closed Riemann surfaces of genus g > 1. Let M be a closed orientable 3-manifold. We will give the following definition according to Zimmermann [18].
Definition 5.1. A closed orientable 3-manifold M is called maximally symmetric if M has a Heegaard splitting of genus g > 1 and a finite group G of orientationpreserving homeomorphisms of maximal possible order 12(g − 1) which preserves both handlebodies of the Heegaards splitting (but does not leave invariant a Heegaard splitting of genus zero or 1).
It was shown that some of well-known 3-manifolds are maximally symmetric, for example, the 3-sphere, the projective 3-space, the 3-torus, the Poincaré homology 3sphere and the Seifert-Weber hyperbolic dodecahedral space. It is also proven that an irreducible maximally symmetric 3-manifold is hyperbolic or Seifert fibred.
Let us consider an orbifold with underlying space S 3 whose singular set is isomorphic to the spatial graph with four vertices pictured in Figure 5.1, where σ denotes a 3-strings braid and 3, m, n are branch indices of corresponding edges with m, n ∈ {2, 3, 4, 5} and indices of other edges are equal 2. Following [18], we denote this orbifold by θ(σ ,m,n).
We will show that the 3-manifold M( , 3) is also maximally symmetric for all ≥ 3 using the following nice criterion from [18].
Let be odd. Then the manifold M( , 3) can be obtained as a 3-fold covering of the 3-sphere branched over the knot ( + 1) 1 and the orbifold ( + 1) 1 (3) has a rotation symmetry of order two denoted by τ.   Thus the quotient space b(4 −2, )(2, 3) of the orbifold ( +1) 1 (3) by the involution τ is an orbifold whose underlying space is the 3-sphere S 3 and whose singular set is two component link b(4 −2, ). Moreover b(4 −2, )(2, 3) has an involution σ whose axis intersects the singular set of b(4 − 2, ) (2,3) in four points (see Figure 5.2a). The quotient space b(4 − 2, )(2, 3)/ σ by the involution σ is an orbifold whose underlying space is the 3-sphere S 3 and whose singular set is a spatial graph with four vertices, pictured in Figures 5.2 and 5.3, that has one edge with branch index 3 and the other edges with branch indices 2. For the case when is even, we can apply similar arguments.